I currently have three websites hosted by one web hosting company. When I purchase Movable Type, does the webhost install the software just once for me or do they have to install it three times, once for each website? Each website has its own database.

Does one purchase allow me to do this?

Each license covers a single installation on a single server.

Once you purchase your license, you can log into Your Account and open a help ticket to discuss the particulars of your hosting situation with support staff, who will work with you on the best approach for your multiple domains.

I'm in a similar situation, but am trying to convince a client to buy an MT commercial license for a project.

On the same server, is it possible to use 1 license across multiple domain names? In their case, they're hosting 17 sites (it's a group of Japanese colleges) on the same server, each with a different URL. The difference between 1 and 17 licenses is naturally large enough to make or break the proposal, and buying 1 license to be able to open a help ticket isn't an option. Any help greatly apreciated. Thanks.

I'd suggest you use the Contact Form to ask your question. You'll want to be as specific as you can about the details.